Question

the point b lies on the segment $overline{ac}$. find the coordinates of b so that the ratio of ab to bc is 7 to 2. a (-3, 4) b (?,?) c (24, -32) coordinates of b : (, )

Explanation:

Step1: Recall section - formula

The formula for finding the coordinates of a point \(B(x,y)\) that divides the line - segment joining \(A(x_1,y_1)\) and \(C(x_2,y_2)\) in the ratio \(m:n\) is \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here, \(x_1=-3,y_1 = 4,x_2=24,y_2=-32,m = 7,n = 2\).

Step2: Calculate the x - coordinate of B

\[

$$\begin{align*} x&=\frac{7\times24+2\times(-3)}{7 + 2}\\ &=\frac{168-6}{9}\\ &=\frac{162}{9}\\ &=18 \end{align*}$$

\]

Step3: Calculate the y - coordinate of B

\[

$$\begin{align*} y&=\frac{7\times(-32)+2\times4}{7 + 2}\\ &=\frac{-224 + 8}{9}\\ &=\frac{-216}{9}\\ &=-24 \end{align*}$$

\]

Answer:

\((18,-24)\)